Geometric control theory uses geometric methods to control various mechanical systems, [1]}, [2]}. We use methods of sub–Riemannian geometry and Hamiltonian concept, [3]}, [4]}. As a reasonable starting point, we consider mechanisms moving in the plane, typically wheeled mechanisms like cars, cars with trailers, robotic snakes, etc., see e.g. [5]}, [6]}. The movement of planar mechanisms is always invariant with respect to the action of the Euclidean group \(SE(2)\) . As prototypes of planar mechanisms we choose those consisting of a body in the shape of a triangle and three legs connected to the vertices of the body by joints which can be of various types and combinations. Although such mechanisms have almost the same shape, the configuration spaces may be very different. In particular, possible motions of the mechanism induce a specific filtration in the configuration space. We present two examples that carry the filtration \((3,6)\)  and \((4,7),\)  respectively, [6]}, [8]}.
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